Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133271 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8524 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,4,8,17,26,6,11,20,29,18,9,21,27)(3,12,14,30,15,7,22,24,16,23,31,28,13,25,19,32);(1,3)(2,7)(4,13)(5,19)(6,16)(8,12)(9,23)(10,28)(11,25)(14,21)(15,18)(17,31)(20,22)(24,26)(27,32)(29,30);(2,8,11,9)(5,20)(7,12,25,23)(10,26,29,27)(14,31)(17,21)(19,22)(24,30,32,28)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,f^c=fhi,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72292717, C2×#133271 |
| maximal normal subgroup of | #72292717, C2×#133271 |
| maximal quotient of | C2×#133271, #72292832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40467);